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LCM of 45 and 120

LCM of 45 and 120 is the smallest number among all common multiples of 45 and 120. The first few multiples of 45 and 120 are (45, 90, 135, 180, 225, 270, . . . ) and (120, 240, 360, 480, 600, 720, . . . ) respectively. There are 3 commonly used methods to find LCM of 45 and 120 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 45 and 120
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 45 and 120?

Answer: LCM of 45 and 120 is 360.

Explanation:

The LCM of two non-zero integers, x(45) and y(120), is the smallest positive integer m(360) that is divisible by both x(45) and y(120) without any remainder.

Methods to Find LCM of 45 and 120

The methods to find the LCM of 45 and 120 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 45 and 120 by Listing Multiples

To calculate the LCM of 45 and 120 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 45 (45, 90, 135, 180, 225, 270, . . . ) and 120 (120, 240, 360, 480, 600, 720, . . . . )
Step 2: The common multiples from the multiples of 45 and 120 are 360, 720, . . .
Step 3: The smallest common multiple of 45 and 120 is 360.

∴ The least common multiple of 45 and 120 = 360.

LCM of 45 and 120 by Prime Factorization

Prime factorization of 45 and 120 is (3 × 3 × 5) = 3² × 5¹ and (2 × 2 × 2 × 3 × 5) = 2³ × 3¹ × 5¹ respectively. LCM of 45 and 120 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ = 360.
Hence, the LCM of 45 and 120 by prime factorization is 360.

LCM of 45 and 120 by Division Method

To calculate the LCM of 45 and 120 by the division method, we will divide the numbers(45, 120) by their prime factors (preferably common). The product of these divisors gives the LCM of 45 and 120.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 45 and 120. Write this prime number(2) on the left of the given numbers(45 and 120), separated as per the ladder arrangement.
Step 2: If any of the given numbers (45, 120) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 45 and 120 is the product of all prime numbers on the left, i.e. LCM(45, 120) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.

☛ Also Check:

LCM of 45 and 120 Examples

Example 1: Verify the relationship between GCF and LCM of 45 and 120.

Solution:

The relation between GCF and LCM of 45 and 120 is given as,
LCM(45, 120) × GCF(45, 120) = Product of 45, 120
Prime factorization of 45 and 120 is given as, 45 = (3 × 3 × 5) = 3² × 5¹ and 120 = (2 × 2 × 2 × 3 × 5) = 2³ × 3¹ × 5¹
LCM(45, 120) = 360
GCF(45, 120) = 15
LHS = LCM(45, 120) × GCF(45, 120) = 360 × 15 = 5400
RHS = Product of 45, 120 = 45 × 120 = 5400
⇒ LHS = RHS = 5400
Hence, verified.
Example 2: The GCD and LCM of two numbers are 15 and 360 respectively. If one number is 120, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 120 × y
⇒ y = (GCD × LCM)/120
⇒ y = (15 × 360)/120
⇒ y = 45
Therefore, the other number is 45.
Example 3: The product of two numbers is 5400. If their GCD is 15, what is their LCM?

Solution:

Given: GCD = 15
product of numbers = 5400
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 5400/15
Therefore, the LCM is 360.
The probable combination for the given case is LCM(45, 120) = 360.

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FAQs on LCM of 45 and 120

What is the LCM of 45 and 120?

The LCM of 45 and 120 is 360. To find the LCM (least common multiple) of 45 and 120, we need to find the multiples of 45 and 120 (multiples of 45 = 45, 90, 135, 180 . . . . 360; multiples of 120 = 120, 240, 360, 480) and choose the smallest multiple that is exactly divisible by 45 and 120, i.e., 360.

How to Find the LCM of 45 and 120 by Prime Factorization?

To find the LCM of 45 and 120 using prime factorization, we will find the prime factors, (45 = 3 × 3 × 5) and (120 = 2 × 2 × 2 × 3 × 5). LCM of 45 and 120 is the product of prime factors raised to their respective highest exponent among the numbers 45 and 120.
⇒ LCM of 45, 120 = 2³ × 3² × 5¹ = 360.

Which of the following is the LCM of 45 and 120? 15, 32, 50, 360

The value of LCM of 45, 120 is the smallest common multiple of 45 and 120. The number satisfying the given condition is 360.

If the LCM of 120 and 45 is 360, Find its GCF.

LCM(120, 45) × GCF(120, 45) = 120 × 45
Since the LCM of 120 and 45 = 360
⇒ 360 × GCF(120, 45) = 5400
Therefore, the greatest common factor (GCF) = 5400/360 = 15.

What is the Least Perfect Square Divisible by 45 and 120?

The least number divisible by 45 and 120 = LCM(45, 120)
LCM of 45 and 120 = 2 × 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 2, 5]
⇒ Least perfect square divisible by each 45 and 120 = LCM(45, 120) × 2 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

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